$$
\begin{aligned}
P(H \mid E) & =\frac{P(E \mid H) \cdot P(H)}{P(E)} \\
p(\boldsymbol{X} \mid \alpha) & =\int_{\theta} p(\boldsymbol{X} \mid \theta) p(\theta \mid \alpha) \mathrm{d} \theta \\
p(\theta \mid \boldsymbol{X}, \alpha) & =\frac{p(\boldsymbol{X} \mid \theta) p(\theta \mid \alpha)}{p(\boldsymbol{X} \mid \alpha)} \propto p(\boldsymbol{X} \mid \theta) p(\theta \mid \alpha)
\end{aligned}
$$